The Bay area is one of the most expensive places in the world though. There are plenty of areas in this country where $15/hour suddenly gives you close to a median income.
Ah, but the reason that the Bay Area is so expensive is that there's such a shortage of land to build housing. In places where the median income is much lower, yes you'll increase demand for housing, but the market will absorb it easily, because there's no shortage of land.
For example, Nashville has a median per-capita income of about $29k. That's just slightly under $15/hour, in theory, ignoring the impact of children on the per-capita numbers. Housing in Nashville is relatively cheap (compared to the Bay Area) at $1053/month.
What would happen if Nashville raised its minimum wage to $15/hr.? Well, a lot of people would have more money to spend. Some percentage of them would spend some percentage of that money on better housing.
Now most people in Nashville aren't having to do apartment sharing eight ways just to make ends meet. The exceptions are mostly college students and new renters, who are just a tiny percentage of the market (unlike in the Bay Area with its staggering rents). So the number of new housing units required would be a fairly small percentage of the market.
The bigger impact would be from people moving up to higher grades of housing or larger apartments with more rooms. In theory, this would drive the price of higher-end housing up. But what happens to all the low-rent housing? Suddenly, you have landlords who can't rent their rooms. So they will spend money to bring their apartments up to higher standards so that they can charge higher fees and bring in people who have more money. They break even, the price of the absolute cheapest housing goes up, and the quality of the housing goes up to match. More importantly, the number of housing units at that higher class goes up, balancing out the increase in demand, so the price for that class of housing actually remains about the same.
So it would have an impact, just not a very big one, and mostly at or near the very bottom of the housing market cost-wise.
Yeah but I don't think you can assume that if you're artificially raising salaries. With respect to education, this raise will affect many people who are already done with education, as well as people who simply aren't capable of finishing high school or going to college.
Yes, which means that there could be some short-term impact, but over the longer term, the trend should reverse, at least in theory.
But even if we assume artificially raising salaries will lead to the same results, it becomes question of whether more people will be bumped from the very poorest into a lower birthrate bucket, or from the $20k range into the $30k range which would result in an increase.
That's a very good question, and I suspect that the answer is "Nobody knows for sure." :-)